Distances between spectral densities. V x. V y. Genesis of this talk. The key point : the value of chordal distances. Spectrum approximation problem

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1 Genesis of this talk Distances between spectral densities The shortest distance between two points is always under construction. (R. McClanahan) R. Sepulchre -- University of Cambridge Celebrating Anders Lindquist 75th Birthday Stockholm, November : Spectrum approximation problem (Byrnes, Georgiou, Lindquist) The key point : the value of chordal distances V x Today s talk : based on Sounds like distance between two systems, but with a new twist V y Example: how to compute the distance between two lines in the plane? 3 4

2 The classical answer: chordal distance Unitary chordal distances are popular in approximation problems V x e.g. planar rotations: e i sin( ) ˆx ŷ V y x e i Intrinsic distances between subspaces can be expressed in terms of principal angles. Unitary chordal distances replace the angles with their sinus. They retain the invariance by rotation. k ˆx k =k ŷ k =1 A property at the core of matrix approximation problems in engineering identify the lines with unit vectors and compute k ˆx ŷ k 5 6 The gap metric is a chordal distance between LTI systems Gap metrics are computed via H-infty norms of transfer functions The gap metric is a distance between subspaces (graphs of pairs (u,y)=gw). To be computable, the distance should be a chordal distance. A chordal distance will be invariant by rotation (unitary transformations) provided that the subspaces are images of unitary operators. 7 8

3 An alternative answer: log chordal distance V x What is a good distance between rational spectral densities? (x, 1 x ) (y, 1 y ) V y Our hint: log chordal d(v x, V y ) = max(log x y, log y x ) 1. Stochastic LTI systems are represented by positive rather than unitary operators. Computable distances are chordal. Hence : what is the log-chordal distance between spectral densities? scale-invariant as opposed to rotation-invariant 9 10 Outline Log chordal distances in cones 1. Log chordal distances in cones. Application to the cone of spectral densities 3. Desirable properties of a distance 4. Comparison with other distances Thompson (or part) metric (196) A close cousin of Hilbert metric 11 1

4 Application to the cone of spectral densities Proof: Desirable properties of a distance Scale invariance of the distance Computable Invariant optimisable Congruence (or filtering) invariance: T R n n [z] Using an invariant distance in approximation problems makes the solution unaffected by filtering the data A source of robustness in modeling! Invariant properties are the main source of non-euclidean geometries 15 16

5 Differential geometry of log chordal distances Outline norm Thompson metric endows the cone with a Finsler manifold structure (similar with Riemannian structure but the norm in the tangent space does not derive from an inner product). length 1. Log chordal distances in cones. Application to the cone of spectral densities 3. Desirable properties of a distance 4. Comparison with other distances log-chordal geodesic The monovariate case The static case (distance on the SDP cone) d T (,I) = log max( M, 1 m ) s 1 d( 1, ) = s 1 d( 1, ) = Z Z (log 1 to be compared with Z ) d ( 1 (D 1 log 1 ) d (=k D 1 1 log log( 1 )d ) (Georgiou, 006) k ) (Martin, 005) to be compared with d(,i)=k log k F = q X log i (This is the Fisher-Rao metric, see e.g. Smith 005) Scale invariance implies a measure of distortion. The main difference lies in the choice of the two versus infinite norm. 19 Invariance implies a logarithmic measure of spectral quantities. The main difference lies in the choice of the two versus infinite norm. 0

6 The general case Outline to be compared with 1. Log chordal distances in cones ( two norm Riemannian analog). Application to the cone of spectral densities or 3. Desirable properties of a distance (divergence measure) 4. Comparison with other distances (Jian, Ning, Georgiou 01) Thompson metric combines the geometrical properties of the two norm with the computational properties of the divergence measures. 1 Conclusions Distances in cones 1. Thompson metric in the cone of spectral densities enjoys a number of desirable properties. The underlying geometry of cones is Finslerian rather than Riemannian 3. A new avenue for distances between systems with a conic representation, e.g. gaussian processes and passive systems An overwhelming topic in Information geometry Convex analysis Optimization Optimal transport Theory of monotone operators Differential geometry An overwhelming number of applications in system theory Covariance matrices Gaussian distributions probability vectors Monotone systems Consensus theory Kalman filtering Spectral estimation Quantum estimation and control The Magritte picture is perhaps not entirely right here 3 4